A formal proof in Coq of a control function for the inverted pendulum

Rouhling, Damien

doi:10.1145/3167101

Cited by 12 publications

(11 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Examples in PVS include semantic invariant reasoning with hybrid automata [1] and, after submission of this article and publication of its precursor [28], dL-style verification with semialgebraic sets and real analytic functions [59]. An earlier formalisation of the control function of an inverted pendulum [57] uses the Coquelicot library. Also in Coq, the robot operating system (ROSCoq) framework uses a shallowly embedded logic of events to reason about hybrid systems but only with dL's differential induction rule.…”

Section: Related Workmentioning

confidence: 99%

Predicate Transformer Semantics for Hybrid Systems

Munive

Struth

2021

J Autom Reasoning

View full text Add to dashboard Cite

We present a semantic framework for the deductive verification of hybrid systems with Isabelle/HOL. It supports reasoning about the temporal evolutions of hybrid programs in the style of differential dynamic logic modelled by flows or invariant sets for vector fields. We introduce the semantic foundations of this framework and summarise their Isabelle formalisation as well as the resulting verification components. A series of simple examples shows our approach at work.

show abstract

Section: Related Workmentioning

confidence: 99%

Predicate Transformer Semantics for Hybrid Systems

Munive

Struth

2021

J Autom Reasoning

View full text Add to dashboard Cite

show abstract

“…New stability proof rules like GLyap can also be soundly and syntactically justified in dL without the need for (low-level) semantic reasoning about the underlying ODE mathematics. As an example of the latter, semantic approach, LaSalle's invariance principle is formalized in Coq [7] and used to verify the correctness of an inverted pendulum controller [32].…”

Section: Related Workmentioning

confidence: 99%

Deductive Stability Proofs for Ordinary Differential Equations

Tan

Platzer

2021

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Stability is required for real world controlled systems as it ensures that those systems can tolerate small, real world perturbations around their desired operating states. This paper shows how stability for continuous systems modeled by ordinary differential equations (ODEs) can be formally verified in differential dynamic logic (). The key insight is to specify ODE stability by suitably nesting the dynamic modalities of with first-order logic quantifiers. Elucidating the logical structure of stability properties in this way has three key benefits: i) it provides a flexible means of formally specifying various stability properties of interest, ii) it yields rigorous proofs of those stability properties from ’s axioms with ’s ODE safety and liveness proof principles, and iii) it enables formal analysis of the relationships between various stability properties which, in turn, inform proofs of those properties. These benefits are put into practice through an implementation of stability proofs for several examples in KeYmaera X, a hybrid systems theorem prover based on .

show abstract

“…Cohen and Rouhling [8] formalized LaSalle's invariance principle in the Coq proof assistant. This formalization was later used by Rouhling to formalize the correctness of a controller for the inverted pendulum [29]. LaSalle's principle uses properties of the ω-limit set; all of the required properties have been formalized in our work (Section 3).…”

Section: Related Workmentioning

confidence: 99%

The Poincaré-Bendixson theorem in Isabelle/HOL

Immler

Tan

2020

Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs

View full text Add to dashboard Cite

The Poincaré-Bendixson theorem is a classical result in the study of (continuous) dynamical systems. Colloquially, it restricts the possible behaviors of planar dynamical systems: such systems cannot be chaotic. In practice, it is a useful tool for proving the existence of (limiting) periodic behavior in planar systems. The theorem is an interesting and challenging benchmark for formalized mathematics because proofs in the literature rely on geometric sketches and only hint at symmetric cases. It also requires a substantial background of mathematical theories, e.g., the Jordan curve theorem, real analysis, ordinary differential equations, and limiting (long-term) behavior of dynamical systems. We present a proof of the theorem in Isabelle/HOL and highlight the main challenges, which include: i) combining large and independently developed mathematical libraries, namely the Jordan curve theorem and ordinary differential equations, ii) formalizing fundamental concepts for the study of dynamical systems, namely the α, ω-limit sets, and periodic orbits, iii) providing formally rigorous arguments for the geometric sketches paramount in the literature, and iv) managing the complexity of our formalization throughout the proof, e.g., appropriately handling symmetric cases. CCS Concepts • Mathematics of computing → Ordinary differential equations; • Theory of computation → Logic and verification.

show abstract

A formal proof in Coq of a control function for the inverted pendulum

Cited by 12 publications

References 22 publications

Predicate Transformer Semantics for Hybrid Systems

Predicate Transformer Semantics for Hybrid Systems

Deductive Stability Proofs for Ordinary Differential Equations

The Poincaré-Bendixson theorem in Isabelle/HOL

Contact Info

Product

Resources

About